1: verify that x = ce−2t is a solution to x 0 = −2x. find c to solve for the initial condition x(0) = 100

The value of tanФ = -1/4 in the III quadrant is -14⁰:

A trigonometric ratio is a ratio between two sides of a right triangle. The tangent ratio is just one of these ratios. In this tutorial, you'll see how to find the tangent of a particular angle in a right triangle.

Tangent = opposite /Adjacent

Tan ∅ = -1/4

Tan ∅= -0.25

Taking the tangent inverse to get ∅

∅=Tan⁻0.25

∅=-14

This is to say that in the III quadrant, Tan ∅=-1/4 is -14⁰

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The radius of each sphere is 0.63 cm.

What is Sphere?

It is given that 15 solid spheres are made by melting a solid metallic cone.

The base diameter (D) of  solid metallic cone is 2 cm and its height (H) is 15 cm.

So, the radius (R) of the solid metallic cone is 1 cm.

Now, volume of the cone (V) can be given as,

[tex]V = \frac{1}{3} \pi R^{2} H\\\implies V = \frac{1}{3} \pi \times 1^{2} \times 15[/tex]

Let the radius of the solid sphere be r.

Then, the volume of one sphere (V') can be given as,

[tex]V'=\frac{4}{3}\pi r^{3}[/tex]

The volume of 15 spheres can be given as,

[tex]V' \times 15=\frac{4}{3}\pi r^{3} \times 15[/tex]

Here, Volume of solid metallic cone = Volume of 15 solid spheres

[tex]\implies V = 15 \times V'\\\implies \frac{1}{3} \pi \times 1^{2} \times 15 = 15 \times \frac{4}{3}\pi r^{3}[/tex]

Solving for r, we get

[tex]r^{3} =\frac{1}{4}\\ \implies r=\sqrt[3]{\frac{1}{4}} \\\implies r= 0.63[/tex]

Therefore, the radius of each sphere is 0.63 cm.

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According to the information, the possible combinations that Jane can have are: Coffee and waffle, Coffee and pancakes, coffee and toast, mocha and waffle, mocha and pancakes, mocha and toast, latte and waffles, latte and coffee, and latte and toasts. In total there are 9 options.

To identify the possible options that Jane has, we must identify the products that are available and the combination that she wants. In this case, it is a hot drink and a bakery product.

According to the above, the possible options would be:

In total there are 9 options.

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Answer:

y = 1/4x + 8

Step-by-step explanation:

-x+4y=32

4y = x + 32

y = 1/4x + 8

The solution to sin(t) is sin(t) = -√[1 -  cos²(t)]

From the question, we have the following parameters that can be used in our computation:

The angle t is the quadrant IV

This means that

sin²(t) + cos²(t) = 1

Subtract cos²(t) from both sides of the equation

So, we have the following representation

sin²(t) = 1 -  cos²(t)

Take the square root of both sides

sin(t) = ±√[1 -  cos²(t)]

sin(t) is negative in quadrant IV

So, we have

sin(t) = -√[1 -  cos²(t)]

Hence, the solution is sin(t) = -√[1 -  cos²(t)]

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The total number of people are 1064 to whom the single cell phone towers are assigned in a particular city.

In a graphical or tabular format, a frequency distribution displays the frequency of repeated entries. It provides a visual representation of the frequency of various items or counts their occurrences.

There are total of 44,700 people in the city and total of 42 cell ,

so to find out the number of people using single cell tower is,

Number of people on single cell tower = 44700/42

=1064.285 ≈ 1064 people

The gathered information is arranged in tables using the frequency distribution method. The information might include student grades, local weather information, volleyball match point totals, etc. We must provide data in a relevant way for improved understanding after data gathering. Create a table that contains a summary of all the data's characteristics. The term "frequency distribution" refers to this.

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In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. We can write the values of AB and BC as -AB = 6 units,AC = 8 units

Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.

Given is that -

{B} is between {A} and {C}. Also : AB = 2x, BC = 3x - 1, AC = 14.

We can write -

AC = AB + BC

14 = 2x + 3x - 1

5x = 15

x = 3

So -

AB = 2 x 3 = 6

AC = 3 x 3 - 1 = 8

Therefore, we can write the values of AB and BC as -

AB = 6 units

AC = 8 units

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The toy cars Anna must sell for her revenue to equal her expense is 4 toy cars. The result is obtained by using the concept of calculating algebra.

To calculate the algebra with one variable, sum up all the same variable and count the value.

Anna want to sell toy car at a craft fair. Each car is sold at a price of $ 12. The material for each car cost $ 3.50 and the table rental at the fair is 34 dollar for the day.

Find the toy cars must Anna sell for her revenue to equal her expense!

We have

Let's say

Revenue = Expense

a × $12 = (a × $3.5) + $34

12a = 3.5a + 34

12a - 3.5a = 34

8.5a = 34

a = 4 toy cars

Hence, Anna must sell 4 toy cars so that her revenue is equal to expense.

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Answer: It could be 7 or 10 or 113

Step-by-step explanation:  Just to make sure that you don't think it's only for 5. If I just get something, that something is equal to itself, which is just going to be true no matter what x you pick.

Pythagoras theorem states that in a right angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. We can represent the length of the hypotenuse as "c", and the lengths of the other two sides as "a" and "b". So the equation is:

c² = a² + b²

To form an equation in x we can use a and b as variable, so we can substitute a = x and b = x-3, the equation becomes:

c² = x² + (x-3)²

To solve for c we can expand the second square and get:

c² = x² + x² - 6x + 9

Now we can add x² on both sides and get:

x² + x² = 6x - 9 + c²

We can combine like terms and get:

2x² = 6x - 9 + c²

Now we can move all the terms to one side and get:

2x² - 6x + 9 - c² = 0

Now we have an equation of a quadratic form with variable x, which can be solved using the quadratic formula.

x² = (6/2)x - 9/2 = 3x - 9/2 = 3x - 4.5 = 0

It can also be factored to get the solutions x = 4.5, x = 0

So the solutions to this equation are x = 4.5, x = 0.

A perfectly competitive firm will produce 10 units of output at a market price of $50 by setting its marginal cost (SRMC) equal to the market price: 2q + 30 = 50. The short-run supply function of this firm is q = 10 when p = $50.

a. If the market price is $50, this firm will produce the output level at which the marginal cost (SRMC) equals the market price. To find this output level, we set the marginal cost equal to the market price:

2q + 30 = 50

2q = 20

q = 10

So the firm will produce 10 units of output at a market price of $50.

It is not necessarily true that $50 is a long-run equilibrium price. In the long run, a perfectly competitive firm will exit the market if the market price is lower than its average total cost, and it will enter the market if the market price is higher than its average total cost. Therefore, to determine if $50 is a long-run equilibrium price, we need to know the firm's average total cost.

b. The short-run supply function of a perfectly competitive firm is the relationship between the market price and the quantity of output that the firm is willing and able to produce. It is found by combining the cost function (SRTC) and the profit-maximizing rule (producing at the level where marginal cost equals the market price).

The equation of the short-run supply function is:

q = (50 - 30)/2 = 10 units of output at a market price of $50

So the short-run supply function of this firm is:

q = 10 when p = $50

Complete Question:

A perfectly competitive firm has the following short-run cost function: SRTC = q2+30q+400

The corresponding short-run marginal cost function is given by: SRMC = 2q + 30

a. If the market price is $50, how much output will this firm produce? Is $50 a long-run equlibrium price? Explain your reasoning.

b. Find the equation of this firm's short-run supply function.

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For the given Poison variables, the probability that X = 2 is 0.22.

Poisson variables are used to model count data, such as the number of customers arriving at a store in an hour, or the number of goals scored in a soccer match. The mean and variance of a Poisson variable are equal.

In this problem, you are given two independent Poisson variables, X and Y, where X has a mean of 3 and Y has a mean of 4. The sum of these two variables is 7. Your goal is to find the probability that X = 2.

Here we need to solve this problem, we can use the formula for the Poisson distribution. The probability of X = k is given by:

=>[tex]P(X = k) = \frac{(e^{-\mu}) * (\mu^k)}{k!}[/tex]

where μ is the mean of the Poisson variable, and e is the mathematical constant approximately equal to 2.718.

Now by substituting the values for X, we have:

=> [tex]P(X = 2) = \frac{(e^{-3}) * (3^2)}{2!}[/tex]

=> P(X = 2) =  (0.0498) * (9) / 2

=> 0.224

Therefore the correct option is D) 0.22.

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The series is the converge series and the sum of the converges is 55/16

The term converge series in math is defined as a series is the sum of the terms of an infinite sequence of numbers

Here we have know that the series 20/8 + 80/64 − 320/512 + 1280/4096.

When we simplify this series the  we get,

=> 20/8 + 80/64 − 320/512 + 1280/4096

=> 5/2 + 5/4 - 5/8 + 5/16

As per the following rule If ∑an is conditionally convergent and r is any real number then there is a rearrangement of ∑an whose value will be r.

Based on these we have to simplify then we get, here with unlike denominators find the Least Common Denominator (LCD)

=> LCD = 16

Multiplying numerators and denominators to get the LCD in all fraction denominators

Then we get,

=> 40/16 + 20/16 - 10/16 + 5/16

=> 55/16

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Using simultaneous equations, the investor invested $7,560 in the 2% term, if the total investment averaged 3.75% for the year.

Simultaneous equations are a system of equations solved concurrently.

The solutions to a system of equations are found at the same time.

The total investment in the portfolio = $42,000

Investment in a 2% term for one year = a

Investment in 4.5% mortgage for one year = b

The average earnings from the investment = 3.75%

a + b = 42,000... Equation 1

(0.02a + 0.045b) ÷ 2 = 0.0375

0.01a + 0.0225b = 0.0375... Equation 2

Multiply Equation 1 by 0.0225:

0.0225a + 0.0225b = 945 ... Equation 3

Subtract Equation 2 from Equation 3:

0.0225a + 0.0225b = 945

-

0.01a + 0.0225b = 0.0375

0.0125a = 944.9625

a = 7,560

= $7,560

b = 42,000 - 7,560

b = 34,440

= $34,440

Thus, the investor invested $7,560 in the 2% term and $34,440 in the mortgage.

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(E) 7/6 π-√3/2

We divided the white region into 5/6 of a circle with radius $1$ and two equilateral triangles with side length $1$ by drawing four lines from the intersection of the semicircles to their centers. This gives the white region a size of 5/6 π+2. √3/4=5/6π+√3/2. The area of the shaded region is equal to the area of the white region minus the area of the large semicircle. This is equal to 2 π-(5/6 π+√3/2)=7/6 π-√3/2.

 

Note

First, it is 5/6 of a circle because the middle sector has a degree of 180-2.60=60, resulting in 60/360=1/6 of a circle.

The other two each have an area of 180-60/360=1/3 of a triangle

Therefore, the total fraction of the circle IS 1/6+2.1/3=1/6+4/6=5/6


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Answer:

(12,7) and (16,7)

There is no slope as both points are on the same line. So the equation would be:

y=7

Step-by-step explanation:

Check the picture below.

so if we pluck the cylinder from the inside, what's leftover is that cyan ring with a height of 10, well, let's get the area of the ring and simply multiply it by its height.

[tex]\textit{area of a circular ring}\\\\ A=\pi (R^2 - r^2) ~~ \begin{cases} R=\stackrel{outer}{radius}\\ r=\stackrel{inner}{radius}\\[-0.5em] \hrulefill\\ R=15\\ r=11 \end{cases}\implies A=\pi (15^2-11^2)\implies A=104\pi \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{volume of a hollow cylinder}}{104\pi }\cdot 10 ~~ \approx ~~ \text{\LARGE 3267.26}~in^3 ~~ \approx 3266~in^3[/tex]

On solving the provided question, we can say that the linear equation is 7x + 4y = 25.25 => x - 28 = -25.25 and cost is $35.43

A linear equation is one that has the form y=mx+b in algebra. B is the slope, and m is the y-intercept. It's usual to refer to the previous clause as a "linear equation with two variables" because y and x are variables. The two-variable linear equations known as bivariate linear equations. There are several instances of linear equations: 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, and 3x - y + z = 3. It is referred to as being linear when an equation has the form y=mx+b, where m stands for the slope and b for the y-intercept.When an equation has the formula y=mx+b, with m denoting the slope and b the y-intercept, it is referred to as being linear.

Let x and Y represent the unknowable costs of a single hot dog and soft drink, respectively. We require two equations in order to solve for the two unknowns.

Equation 1 says that it costs $35 to buy 10 hot dogs and 5 soft drinks, or 10x + 5y = 35.

Equation 2: A total of $7.25 is spent on 7 hot dogs and 4 soft beverages, or 7x + 4y = $7.25.

[tex]10x + 5y = 35\\5y = 35 - 10x \\y = 7 - 2x \\[/tex]

[tex]7x + 4y = 25.25\\7x + 4(7-2x) = 25.25\\7x + 28 - 8x = 25.25\\-x + 28 = 25.25\\x - 28 = -25.25[/tex]

cost is $35.43

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The weight of the draft horse is [tex]10^6[/tex] times the weight of the paper clip.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

Weight of the paper clip = [tex]10^{-3}[/tex]

Weight of the draft horse = 10³

Now,

Weight of the draft horse.
= [tex]10^6[/tex] x Weight of the paper clip

= [tex]10^6[/tex] x [tex]10^{-3}[/tex]

= [tex]10^6[/tex]  x 1/10³

= 10³

Thus,

The weight of the draft horse is [tex]10^6[/tex] times the weight of the paper clip.

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Answer:

the answer is 6

It will take 70 days for the population of bacteria to reach 10,000 from 1000, if it is growing according to the Malthusian model and doubling itself in 10 days.

This can be calculated using the exponential growth formula: N(t) = N0 * (1 + r)^t, where N0 is the initial population size (1000), r is the growth rate (2/10 = 0.2), and t is the time in days (70).

The Malthusian model of population growth is an exponential growth model developed by Thomas Robert Malthus in 1798. It states that the population of a species increases exponentially, doubling every generation, and is limited only by the available resources.

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Correct option is B, the graph represents the inequality y ≤ - x + 9.

Both mathematical expressions that are created by joining two expressions are called inequalities. The equation's two expressions are deemed to be equal when the equals sign (=) is present. The characters >, or indicate that the two expressions are not necessarily equal in an inequality.

We may see the values that an inequality reflects by placing inequality on a number line. We create a straight line and identify the end points with either an open circle or a closed circle in order to represent inequalities on a number line.

For the given graph,

y ≤ - x + 9

Put x = 0

y ≤ 9

The above condition is correct for the given graph.

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The S&P 500 and XYZ Inc. have a positive linear relationship.

The covariance between two variables provides information about the relationship between them, but it does not clearly measure the strength or type of the relationship. A positive covariance means that the variables move in the same direction, while a negative covariance means that the variables move in opposite directions.

In this case, the covariance between the S&P 500 and XYZ Inc. is 6,317.50, which is a positive value. This suggests that there is a positive linear relationship between the two variables, meaning that as the S&P 500 increases, XYZ Inc. also tends to increase, and vice versa. However, to fully understand the relationship between the two variables, it is necessary to calculate other measures, such as the correlation coefficient, which provides a standardized measure of the power and direction of the relationship between two variables.

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The value of x, obtained using the condition of proportional ratio of the lengths of the sides of similar triangles, is x = 11

Similar triangles are triangles that have the same interior angles, such that the three interior angles of one triangle are congruent to the three interior angles of the other, similar triangles, but the lengths of the sides of the triangles may vary.

The triangles ΔQRS and ΔTUV are similar. Based on the location of the obtuse angle ∠R and ∠T, and the longest sides, RS and UV, we get;

[tex]\frac{RS}{UV} = \frac{RQ}{UT}[/tex]

Therefore, we get;

[tex]\frac{54}{36} = \frac{24}{x+5}[/tex]

54 × (x + 5) = 24 × 36

54·x + 54 × 5 = 24 × 36

54·x = 24 × 36 - 54 × 5 = 594

x = 594 ÷ 54 = 11

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well, we know there are 60 minutes in 1 degree, so in 40 minutes that'll be 40/60 degrees or namely 2/3 of a degree.

We know that 180° is π radians, so let's find 43 ⅔ degrees

[tex]\stackrel{mixed}{43\frac{2}{3}}\implies \cfrac{43\cdot 3+2}{3}\implies \stackrel{improper}{\cfrac{131}{3}} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\begin{array}{ccll} degrees&radians\\ \cline{1-2} 180&\pi \\\\ \frac{131}{3}&x \end{array}\implies \cfrac{180}{~~ \frac{ 131}{3 } ~~} = \cfrac{\pi }{x}\implies \cfrac{\frac{180}{1}}{~~ \frac{ 131}{3 } ~~} = \cfrac{\pi }{x} \implies \cfrac{180}{1}\cdot \cfrac{3}{131}=\cfrac{\pi }{x} \\\\\\ \cfrac{540}{131}=\cfrac{\pi }{x}\implies 540x=131\pi \implies x=\cfrac{131\pi }{540}\implies x\approx 0.762~radians[/tex]

Answer:

8-15

8-23

and

15+23

15+8

Step-by-step explanation:

Answer:

The solution is w ≤ 18

It is difficult to predict the exact value of the correlation coefficient between crime rate and educational attainment across metropolitan areas.

However, it is often observed that there is a negative correlation between these two variables, meaning that as the level of educational attainment increases, the crime rate decreases.

There is evidence to suggest that higher levels of education are associated with lower levels of criminal behavior and better job opportunities, which can help to reduce crime in a community.

On the other hand, there are many other factors that can affect crime rates, such as poverty, unemployment, and population density, so the relationship between crime and education is complex and multi-dimensional.

It is also possible that the correlation coefficient between crime rate and educational attainment could be close to zero or even positive in some cases, depending on the specific metropolitan area and the data available.

Therefore, it is not possible to definitively say that the correlation coefficient will be negative, positive, zero, or none of the above.

Further analysis and study would be needed to determine the specific relationship between crime rate and educational attainment in any given metropolitan area.

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The measurement of the angle OP is 39

The given parameters that will help us to answer the question are

R is mid point of OQ

S is mid point of PQ

RS = 3x + 38

OP = 4x- 14

R is mid point of OQ and S is mid point of PQ;

By using mid point theorem

[1/2][OP] = RS

This implies that So,

[1/2][3x + 38] = [4x- 14]

[3x+38] = 2[4x- 14]

Opening the brackets we have

3x+38=8x-28

Collecting like terms

3x-8x=-28-38

-5x=-66

Making x the subject of the relation we have

x = -66 / -5

x = 13.2

Therefore RS = 3(13.2) + 38 = -11.4

OP = 4(13.2)- 14=38.8

Measurement of OP = 39 approximately

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The system of equations has an infinite number of solutions. The correct answer would be an option (D).

The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are equal.

The system of equations is given in the question, as follows:

-5.9x - 3.7y = -2.1     .....(i)

5.9x + 3.7y =  2.1     .....(ii)

As we know that when a solution set of infinite points exists, a system of linear equations has an infinite number of solutions.

Here the system of equations has an infinite number of solutions because both equations are identical.

Hence, the correct answer would be an option (D).

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